Design, Optimization, and Control of Tensegrity Structures

172 Pages·2004·46.619 MB·English

by  Milenko Masic

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3 1822 00943 5132
UNIVERSITY OF CALIFORNIA , SAN DIEGO
DESIGN, OPTIMIZATION, AND CONTROL OF TENSEGRITY
STRUCTURES
A dissertation submitt ed in partial satisfaction of the
requirements for the degree Doctor of Philosophy
m
Engineering Sciences (A erospace Engineering)
by
Milenko Masic
Committ ee in charge:
Professor Robert E. Skelton, Chair
Professor Philip E. Gill
Professor William J. Helton
Professor Thoma s R. Bewley
Professor Raymond de Callafon
June 2004
Copyright
Milenko Masic, 2004
All rights reserved.
The dissertation of Milenko Masic is approved , and
it is acceptable in quality and form for publication on
microfilm:
Professor Robert E. Skelton, Committee Chair
University of California, San Diego
2004
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DEDICATION
To my family
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Contents
Signatur e Page iii
Dedication
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Contents
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List of Figures viii
Acknowledgements
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Curriculum Vitae
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Abstract xiii
1 Introduction 1
1.1 History of tensegrity structures 1
1.2 Motivation for the research . 4
1.3 Summary of the dissertat ion 5
2 Statics of tensegrity structur es 8
2.1 Introd uction ......... . 8
2.2 The tensegrity equilibrium conditions 12
2.3 Shape constra ints . . . . . . . . . . . 18
2.4 Tensegrity structure stability analysis 30
2.5 Tensegrity form-finding examples 35
2.6 Conclusions . . . . . . . . . . . . . . 40
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3 Enabling tools for tensegrity form-finding 42
3.1 Introdu ction . . . . . . . . . . . . . . . . . 42
3.2 Invariant tensegrity geometric transformations 43
3.3 Composition of tensegrity structures ..... 45
3.4 Geometry and equilibrium analysis of some tensegrity modules 53
3.5 Geometry and equilibrium of monohedral modular tensegrity plates 68
3.6 Geometry and equilibrium of class-two tensegrity towers 77
3.7 Conclusions ......... ....... . 78
4 Open-loop control of modular tensegrities 80
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Slowly varying nonlinear systems and open-loop control 81
4.3 Examples . 83
4.4 Conclusions 88
5 Optimal mass-to-stiffness-ratio tensegrity design 90
5.1 Introdu ction . . . . . . . . . 90
5.2 Formulation of the problem 92
5.3 · Nonlinear program formulation 100
5.4 Examples . 103
5.5 Discussion . 112
5.6 Conclusions . 113
6 Joint structure and control design 115
6.1 Introdu ction ......... . . . . 115
6.2 Lumped mass dynamic model of a tensegrity structure . 116
6.3 Linearized dynamic model of the structure . . . . . . . . ·. 118
6.4 Designing the structure for the optimal LQR performance - optimiza-
tion over the prestress cone . 125
6.5 Conclusions . . . . . . . . . . . . . . . . . 136
7 General conclusions and future research 137
7.1 Conclusions . . . 137
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7.2 Future research . 138
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8 Appendices 140
8.A Equivalency of the force density and length-minimization method for
tensegrity form-finding . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.B Symmetry of prestress forces .. .. . ... ... .... . ...... . 144
8.C Open-loop control laws for typical elements of modular tensegrities . 148
Bibliography 152
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List of Figures
2.1 Node and plate constraint . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 A tensegrity cross that admits symmetries with respect to the x and
y-axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 left: Class 1 shell-class tensegrity tower that admits symmetries about
z-axis; right: Connectivity diagram of shell-class tensegrity . . . . . . 21
2.4 Example of a symmetric and asymmetric force distribution for a sym-
metric tensegrity structure . . . . . . . . . . . . . . 26
2.5 Shell class tensegrity stage geometry param eters . . . . . . . . . . . 31
2.6 · Connectivity of the unit of the unstabl e unit plate . . . . . . . . . . 35
2. 7 Initial guess for the plate geometry (left) and computed plate geom-
etry(r ight) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.8 Equilibrium shell-class tensegrity towers with different heights 40
2.9 Shell class tensegrity with tapered stages . . . . . . . . . . . . 41
3.1 An equilibrium elliptical tensegrity cross generated by the similarity
transformation from the equilibrium square configuration . . . . . . . 44
3.2 The sparsity pattern of equilibrium equations for a composition of
two tensegrity structures 46
3.3 Tensegrity plates made connecting three, four and six-bar units 52
3.4 Tensegrity tower with two stages made by composition of two different
tensegrity modules . . . . . . . . . . . . . . . . . . . . . 53
3.5 One-stage shell class module geometry and connectivity . ,55
3.6 Two-stage unit equilibrium . . . . . . . . . . . . . . . . . 61
3. 7 Elliptical unit configuration . . . . . . . . . . . . . . . . 68
3.8 Periodic geometry of a tensegrity plate and associated lattice generators 69
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3.9 Matching relative positions for three-bar units .... 71
3.10 Two different topologies of three-bar modular plates . 74
3.11 Matching overlap . . . . . . . . . . . . . . . . . 75
3.12 Force distribution in then-bar stable unit plate 77
3.13 Module distribution in sparse plate 78
3.14 Composition of tower modules . . . 79
4.1 Deployment of a six-bar unit plate 84
4.2 The deployment of a paraboli c tensegrity structure . . . . . . . . 85
4.3 Positions of the top nodes of the paraboli c tensegrity plate vs. a
quadratic parabola . . . . . . . . . . . . . . 85
4.4 Open-loop deployment of a tensegrity tower . . . . . . . . . . . . . . 86
4.5 Internal transversal wave function . . . . . . . . . . . . . . . . . . . . 87
4.6 Self-propelled tensegrity actuated by applying transversal wave shape
control 88
5.1 A typical sparsity pattern of the Jacobian of the nonlinear constraints
for a symmetric problem ; the general shape constraint are excluded . 103
5.2 Initial vs. optimized tensegrity beam in loaded state , showing de
formation under load. Legend: light gray-slack strings , dark gray-
compressed bars, black-stretched strings ................. 104
5.3 Non-uniqueness of the optimal structure - different optimal structure 105
5.4 Initial vs. optimal aspect ratio, L / d, of symmetric tensegrity beam . 106
5.5 Optimal overlap ratio L L, truncation ratio d/ L and objective func-
0/
/y . . . . . . . . . . . . .
tion fT u vs. material yield strain CJ 107
5.6 Initial vs. optimal tensegrity tower under compressive load . .. . .. 109
5. 7 Optimal number of stages vs. tensegrity beam aspect ratio . . . . . . 110
5.8 Optimal number of stages of tensegrity beam vs. material param eters 111
6.1 Tensegrity structure with three prestr ess modes . . . . . . . . . . . . 123
6.2 Extreme direction of the prestress cone of the structure in Figure 6.1 124
6.3 Combination of the extreme directions of the prestress cone for the
structure in Figure 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4 Initial (up) Vs. optimal (d own) distribution of the prestress in the
tensegrity beam . . . . . . . . . ......... .......... 132
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6.5 Convergence of the algorithm - objective Vs. iteration number . .. . 132
6.6 Impulse response of the initial plant Vs. impulse response of the
optimized plant - effect of the structure optimization on its dynamic
performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.7 Closed-loop impulse response of the initial plant Vs. the same re
sponse for the optimized plant 134
6.8 Optimal controllers for the initial and the optimized plants 135
8.1 Left: Symmetric three-bar module in the configurat ion with a= 1r/4.
Right: Prestressed elements of an asymmetric extreme direction of
the prestress cone. . . . . . . . . . . . . . 144
8.2 Connectivity matrix of the three bar unit ... ..... .... . .. . 145
8.3 Elements of the three-bar module that are prestressed in the two
configurations with the extreme values of a . . . . . . . . . . . . . . . 146
8.4 Elements of the three-bar module that are prestressed in each of the
three extreme direction of the prestress cone when a= 1r/ 3 ...... 147
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